Question

Which of the following products, in a hydrogen atom, are independent of the principal quantum number $n$? The symbols have their usual meanings.

• A. ${\omega }^{2}r$
• B. $\frac{E}{v^2}$
• C. $v^{2}r$
• D. $\frac{E}{r}$

Answer 1

Answer: (B), (C)

$v^{2}r$
$\frac{E}{v^2}$

Angular Momentum Quantisation:
$mvr=\frac{nh}{2\pi }$ ------------(1)

Centripetal force:
$\frac{k\left(e\right)e}{r^2}=\frac{m{v}^2}{r}$ ----------------(2)
$v^{2}r=\frac{k{e}^2}{m}$ (Constant)

Substituting (1) in (2):
$v=\frac{2\pi k{e}^2}{nh}$
$v=\frac{C_1}{n}$

Energy:
$E=13.6\frac{Z^2}{n^2}$ -----------(3)
$\frac{E}{v^2}=const.$